The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Pascals triangle

When we talk about mathematical discoveries certain names are mentioned. These are names like Pascal, Euclid, Fermat, and Euler. These people become our mathematician Heroes. In our eyes, we often believe they pioneered the study. When we hear names like Mo Jing and Yang Hui in western society, most of us probably don’t even think anything of them.

But did you know that many of the great mathematical discoveries made in Western Mathematics were also made by Chinese mathematicians? In fact some mathematical discoveries we attribute to western mathematicians were even made by Chinese mathematicians far before they were discovered in the west.

I bring this up not necessarily to shame western culture, but because I find it fascinating. We have two cultures that really didn’t intermix ideas and traditions, yet it seems that they have made many similar mathematical discoveries. In my opinion these similarities in a way show that two totally different cultures with cast differences still have profound similarities that can unite them.

Also in the great debate of whether math is manmade or discovered I personally believe the similarity between western and Chinese mathematics is a point for Team Discovered. That might be only because I currently am on Team Discovered, though. I believe this is a point for Team Discovered because I feel if two separate cultures that are not trading ideas come up with the same mathematical truths then maybe they discovered them instead of just happened to share the same inventive thoughts. Still maybe this is the exact reason I should join Team Invention and I am just not thinking through my argument all the way.

Let’s talk about some of the similar discoveries in Chinese mathematics and western mathematics. Let’s try to focus on the person behind a concept that both cultures discover/invented. If feasible we should mention when the discovery/invention came about and how. Also how did it influence mathematics and the human race? I won’t focus on it, but you might even want to see if what we discuss puts you on team invention or team discovery for mathematics.

I guess the first Chinese work that I would like to point out is more of a compilation of Chinese works than the work of one individual, but did you know that of book very much like Euclid’s Elements existed in China? This book was the canon of a group of people called Mo Jing. They were the followers of Mozi and the canon contained, among philosophical insights, works on geometry. Mozi was actually a Chinese philosopher, but his teachings inspired his followers to consider mathematics as well. In fact this book contained a definition for a point similar to Euclid’s. To be specific, “a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it.” (http://history.cultural-china.com/en/167H8715H13206.html)

Now, one of the most famous mathematical discoveries is “Pascal’s Triangle.” “Pascal’s Triangle,” is a fascinating work. To describe it you build it from the top down. Put a one at the top. Build the triangle down adding 1 more number in each row. The value of the number below is the sum of the two numbers directly above it. If it is an edge case the number is 1.

Image: Drini and Conrad.Irwin, via Wikimedia Commons.

This discovery, made by Pascal while, through letters, he was exploring probability with Fermat, was also discovered much earlier by a Chinese man named Yang Hui. Even before Yang Hui it was described earlier by Jia Xian in 1100. Yang Hui in his book attributes the triangle to Jia and acknowledged that it was through this triangle that he found square roots as well as cubed roots.

I feel it is also very important that we discuss the book Zhou Bi Suan Jing. This book, which is a collection of anonymous works, contains one of the first proofs of the well-known and widely used Pythagorean Theorem. As a refresher, this a2+b2=c2. Controversy overshadows the actual date of the book which is assumed to be around 1046-256 BCE.

We can clearly see that mathematical ideas are not monopolized by western tradition. In fact, in my studies of Chinese mathematics, I found references Pascal’s Triangle being found in India and Iran. Pascal was a genius, but clearly he was not the original discoverer of the triangle that bears his name. Mathematics is a global study, applied in many ways similarly by many cultures.

Take some time and identify a culture. Make sure it is a culture that is so different from your own that, in a history class, this culture would study completely different things than what you studied. Now take what you know of your culture and the culture you chose and find similarities. Sometimes this can be hard. There are similarities such as in many cultures families eat together, but there are also many differences. What I am saying here is that in many ways math can be one of those similarities. This is neat! Math is as much a western study as it is an eastern study.

So next time you are learning about a western mathematician and how awesome he/she is, take some time and ask yourself if maybe the same ideas were explored by someone else in a different time in a different part of the world. Maybe even look it up. You might be surprised by what you find.

Today we are going to be looking into the different patterns in Pascal’s Triangle. I am not talking about Pascal from the Walt Disney Move Tangled. I am talking about Blaise Pascal, a famous French mathematician and philosopher. He was born on June 19, 1623, in Clermont-Ferrand, France. His mother died we he was just three years old, leaving behind his two sisters, his father, and himself. His father, Etienne Pascal, never remarried; instead he focused on educating his children, especially his son, Blaise Pascal. In 1653, Pascal released Traite du triangle arihmetique, which talked about binomial coefficients. This later became famous and became known as Pascal’s Triangle.

Image: Hersfold (public domain), via Wikimedia Commons.

Even though Blaise Pascal was the one to get all the credit for the triangle, the ancient Chinese actually developed it. The reason why he receives the credit for the triangle is because he discovered the patterns that are in the triangle. It could also be because the Europeans did not know about the previous discovery from China, and we have an Eurocentric math culture so we know it as Pascal’s Triangle. Before we go into the different patterns that he discovered, we are going to review how Pascal’s Triangle is made. It starts with the number one at the very top, this is called row 0. Row 1 consists of 1 and 1 and the next row consists of the numbers 1, 2, and 3. This is determined by adding the two numbers above to the left and to the right to get the coefficients in the row. For example, in row 2 to get the first number we add 0+1=1, 1+1=2, 1+0=1. You will do this process for each of the rows to get the different coefficients.

Now that we know more about what Pascal’s Triangle lets look at the many different patterns that are present in Pascal’s Triangle. The first one is called the Hockey Pattern. If you start at any 1 in the triangle and go diagonally until you choose to stop, the sum of all the numbers in that diagonal is the number just below the last number, ensuring that you are looking at the number below and to the opposite side that the diagonal would have continued. If you start on the left side of the triangle you will go down to the right diagonally, then when you want to stop, you will go below and to the left of the last number to find the sum of the numbers within that diagonal. For the right side, you will go below and to the right of the diagonal for the sum. For example, look at the highlight red to see the pattern. 1+6+21+56=84.

Two more patterns that are present in Pascal’s Triangle are called The Sum of Rows and Prime numbers. In the pattern, The Sum of Rows, the website All You Ever Wanted to Know About Pascal’s Triangle and More states: “The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row.” This is saying if you pick the row 5 the sum of all the numbers in that row would equal 25. Remember that the first row is considered row 0. The next pattern, Prime numbers, states that if the second number in a row is prime, then all the other numbers in that row are divisible by that number.

These are just a few of the patterns that are present in Pascal’s Triangle. There are many more patterns and I encourage you to do some more research and discover all the patterns in Pascal’s Triangle.

Math is something that is found all throughout history.It was used for may different reasons, in many different cultures.What I find interesting is how these different cultures learned some of the same ideas without even having knowledge of the others’ work. These works could be anything from counting systems to Pascal’s triangle.It can also include how one culture passed its knowledge on to another. This makes you wonder how some ideas that were known in western civilization could also be found in Asia.As I was looking into this I found some very interesting facts about mathematics in China. Some small examples of math found in China begin with something called oracle bone scripts: scripts carved into animal bones or turtle shells. These scripts contain some of the oldest records in China.This, like the clay from babylonian times, had many different uses including math.Chinese culture also had something called the six arts: Rites, Music, Archery, Charioteerring, Calligraphy, and Mathematics.Men who excelled in these arts were known as perfect gentlemen.

In China, like in India, one can find the use of a base ten numeral system.This is quite different from the Babylonians, which makes it seem like there must have been some conduit of knowledge between India and China.In China, around 200 BCE, they used something called “rod numerals.”Rod numeral counting is very similar to what we use today.This counting system consisted of digits that ranged from one to nine, as well as 9 more digits to represent the first nine multiples of 10.The numbers one through nine were represented by rods going vertically, while the numbers of the power of 10 were horizontal.This means that every other digit was horizontal while its neighbor was vertical.For example 215 would be represented like this ||—|||||.If one wanted to use a zero you would have to use an empty space.The empty space is also something that can been seen in the Babylonian counting system.As with the Babylonians, a symbol was eventually used for zero.Interestingly enough, before there was a symbol for zero, counting rods included negative numbers. A number being positive or negative depended on its color: black or red.This idea of having negative numbers didn’t come about in another culture until around 620 CE in India.It seems quite apparent that several ideas that originated in China could possibly have been passed on to a neighboring country.

Rod numerals. Image: Gisling, via Wikimedia Commons.

The use of counting rods as a counting system brought about another very interesting mathematical concept, the idea of a decimal system.China first used decimal fractions in the 1st century BCE.Fractions were used like they are today, with one number on top of another.For example, today if you used a faction for one half, it would be written like this: 1/2.Using rod numbers you can do the same thing like this: | / ||.Not only could this be represented as a fraction but it could also be written as a decimal.To do this one would simply write the number out and insert a special character to show where the whole number started.For example, if you wanted to say 3.1213, you would write it as a whole number like this: |||—||—|||.To show where the left side of the decimal starts, you would mark it with a symbol under the number to the left of the decimal point, in this case under the first 3.To me the use of rod numbers is so similar to how we use our numbers today that even the arithmetic that was used can be done easily by someone in our culture.Addition is done almost the same except they would work from left to right.Multiplication and division were used as well. The use of base ten as well as using rod numerals made complicated equations much easier to attain, such as the use of polynomials and even Pascals triangle.

The triangle known as “Pascal’s” in the west, in a Chinese manuscript from 1303 CE. Image: Public domain, via Wikimedia Commons.

Centuries before Pascal, the Chinese knew about Pascal’s triangle.Shen Kuo, a polymathic Chinese scientist was known to have used Pascal’s triangle in the 12th century CE.It appears that knowledge of Pascal’s triangle begins even before this. The first finding of Pascal’s triangle was in ancient India around 200 BCE.We can see that this idea was sprouting around and found evidence in different cultures, from Persia to China and to Europe.This again makes one wonder how this knowledge base was passed around from one culture to another.Lacking historic details, it is hard to see if this idea of Pascal’s triangles was thought up individually or if this concept was somehow passed from one culture to another.

It seems that in all cultures there is a need for counting, which in turn brings about the need for math.The cultural implications can mean that you are a “perfect gentlemen” by having mathematical knowledge, or it could lead a greater knowledge that can be passed on to other cultures.In China, we see that many ideas of numbers and mathematics were thought up on their own without having other culture’s ideas intervening.We can also see that the knowledge that was passed on was able to thrive and turn into something even more intriguing.It is apparent that we can always learn and teach others to help our knowledge grow.